Theorem crnggrpd 20245

Description: A commutative ring is a group. (Contributed by SN, 16-May-2024.)

Hypothesis

Ref	Expression
crngringd.1	⊢ (𝜑 → 𝑅 ∈ CRing)

Assertion

Ref	Expression
crnggrpd	⊢ (𝜑 → 𝑅 ∈ Grp)

Proof of Theorem crnggrpd

Step	Hyp	Ref	Expression
1		crngringd.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
2	1	crngringd 20244	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
3	2	ringgrpd 20240	1 ⊢ (𝜑 → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2107  Grpcgrp 18952  CRingccrg 20232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-nul 5305

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435  df-ring 20233  df-cring 20234

This theorem is referenced by:  fermltlchr  21545  psdmul  22171  psd1  22172  psdpw  22175  ply1chr  22311  ply1fermltlchr  22317  elrgspnsubrunlem1  33252  elrgspnsubrunlem2  33253  erlbr2d  33269  rlocaddval  33273  rloccring  33275  rloc0g  33276  rlocf1  33278  fracerl  33309  evl1deg1  33602  evl1deg2  33603  evl1deg3  33604  ply1dg1rt  33605  irngss  33738  irredminply  33758  algextdeglem4  33762  algextdeglem5  33763  aks6d1c1p3  42112  aks6d1c2lem4  42129  aks6d1c6lem2  42173  aks5lem2  42189  evlsbagval  42581  selvvvval  42600  evlselv  42602  selvadd  42603  prjcrv0  42648